Chap 08

Angles and shapes are all

around us.

While sightseeing in

Sydney, Chris, Nola and

Thomas travel across the

Sydney Harbour Bridge and

notice an array of shapes,

angles and solids.

What types of angles can

you identify in this picture?

What types of triangles and

quadrilaterals can you find?

Are there any parallel lines?

What do you notice about the

angles in the shapes?

This chapter looks at the

properties of various shapes,

angles and solids.

8

77eTHINKING

Geometry

322 M a t h s Q u e s t 8 f o r V i c t o r i a

READY?areyou

Are you ready?Try the questions below. If you have difficulty with any of them, extra help can be

obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon

next to the question on the Maths Quest 8 CD-ROM or ask your teacher for a copy.

Classifying angles

1 State the type of angle in each of the following figures.

a b c

Classifying triangles according to the lengths of their sides

2 Complete the following descriptions of different types of triangles.

a Triangles with _________ equal sides are called equilateral triangles.

b Triangles with exactly _________ equal sides are called isosceles triangles.

c Triangles with _________ equal sides are called scalene triangles.

Complementary angles

3 a Find the complement of 30°.

b Find the size of x in the figure shown at right.

Supplementary angles

4 a Find the supplement of 60°.

b Find the size of x in the figure shown at right.

More angle relationships

5 In each of the following figures, find the size of the pronumeral.

a b

Constructing angles with a protractor

6 Construct the following angles using a protractor.

a 20° b 77° c 122°

8.1

8.2

8.4

40ºx

8.5

70º x

8.10

120º

x

100º

50ºx

8.12

C h a p t e r 8 G e o m e t r y 323

What is geometry?The word geometry comes from the Latin words geo meaning earth and metre meaning

measure. It dates back to the ancient Greek philosophers and mathematicians such as

Plato, Euclid and Pythagoras. They considered geometry as the science of measuring

our Earth and describing its properties. It was Euclid, for example, who proved that

parallel lines never meet.

Geometry today looks at angles, shapes and solids, and investigates their relation-

ships and properties. In this chapter you will investigate triangles, quadrilaterals and

other polygons, and work with rules that the ancient Greeks discovered over 2000 years

ago. You will use angles and their properties to describe and construct shapes and

solids.

TrianglesThe triangle is one of the most com-

monly used shapes in the world.

Looking around, we can see its many

practical uses, from house roofs to

pylons to tents. Triangles give strength

and rigidity to geometric shapes.

Learning about the different types of tri-

angles helps us to understand the design

of many everyday objects.

Naming trianglesTriangles are named according to the lengths of their sides, or the size of their angles.

The table below lists the six names commonly used to describe triangles.

Naming triangles according to the lengths of their sides

Equilateral

triangle

All three sides are equal in length.

(Also, all three angles are 60°.)

Isosceles triangle Exactly two sides are equal in length.

(Also, angles opposite the equal sides

are equal in size.)

Scalene triangle No sides are equal in length.

(Also, no angles are equal in size.)

Cabri Geometry

Classifyingtriangles(sides)


Naming triangles according to the size of their angles

Note: To indicate the sides of equal length and angles of equal size, we use identical

marks. To indicate the angle is right (90°), a small square is used.

Acute-angled

triangle

All angles less than 90°.

Right-angled

triangle

Contains a right angle (90°).

Obtuse-angled

triangle

Contains an obtuse angle (greater than

90°).

Cabri G

eometry

Classifying triangles (angles)

Using side and angle markings where appropriate, draw a triangle that is:

a a scalene triangle

b both right-angled and isosceles.

THINK DRAW

a A scalene triangle has 3 sides of

different size. Nothing is known about

the actual length of sides of this triangle.

Draw a triangle, making the sides of 3

different lengths.

a

b A right-angled triangle contains a right

angle and an isosceles triangle has two

sides of equal length. Draw a triangle

with one angle 90° and two sides equal

in length. Place identical markings on

equal sides and use a small square to

indicate the right angle.

b

1WORKEDExample


Triangles

1 Using side and angle markings where appropriate, use your ruler to draw a triangle

that is:

a isosceles b scalene c right-angled d obtuse-angled.

2 Using side and angle markings where appropriate, use your ruler to draw a triangle

that is:

a both obtuse-angled and isosceles b both acute-angled and equilateral

c both acute-angled and isosceles d both right-angled and scalene.

Select the most appropriate name (or names) for the triangle shown.

A Scalene

B Obtuse-angled

C Acute-angled and equilateral

D Scalene and right-angled

E Isosceles

THINK WRITE

There are three identical markings placed on the

sides of the triangle; therefore, all sides are the same

in length. State the side name of this triangle.

Side name: equilateral

All angles in an equilateral triangle are 60°. State the

angle name of this triangle.

Angle name: acute angled

Select the most appropriate answer from the choices

given.

Answer: C

multiple choice

1

2

3

2WORKEDExample

1. Triangles can be named according to the length of their sides or the size of their

angles.

2. Angle names are:

(a) acute-angled triangle (all angles are acute)

(b) right-angled triangle (contains one right angle)

(c) obtuse-angled triangle (contains one obtuse angle).

3. Side names are:

(a) equilateral (all sides are equal in length)

(b) isosceles (exactly two sides are equal in length)

(c) scalene (all sides are different in length).

remember

8AWORKED

Example

1a

WORKED

Example

1b


3 Copy and complete this table.

Note: Allow enough space to fit in the missing figures.

4 Four types of triangles are marked in

this power transmission tower. Name

each type of triangle by completing

these sentences.

Triangle:

a A is an ___________________

triangle.

b B is a ______________________

triangle.

c C is a ___________________

triangle.

d D is an ___________________

triangle.

5

Select the most appropriate name

(or names) for the triangles shown.

a

Picture Name of triangle Definition

Isosceles triangle

One angle is greater than 90°.

Right-angled triangle

All angles are less than 90°.

A Right-angled

B Right-angled and acute-angled

C Right-angled and scalene

D Right-angled and isosceles

E Right-angled and obtuse-angled

SkillS

HEET 8.1

Classifying angles

SkillS

HEET 8.2

Classifying triangles according to the lengths of their sides

Cabri G

eometry

Classifying triangles (sides)

Cabri G

eometry

Classifying triangles (angles)

Mathc

ad

Classifying triangles

A

B

C

D

WORKED

Example

2

multiple choice


b

c

d

6 The Israeli flag at right is made up of a blue Star of

David on a white background. Which triangles are

used in the Star of David?

7 The set of swings at right is constructed

with metal tubing that makes an acute

angle with the ground. Which type of

triangle is made with the two supports and

the ground (and marked in red)?

8 Using 12 matches, construct each of the

following triangles. Use your ruler to draw

the solutions in your workbook, clearly

showing the number of matches used on

each side.

a An equilateral triangle

b An isosceles triangle

c A scalene triangle

9 Use matches to construct each of the following structures. Use your ruler to draw the

solution in your workbook.

a Use 9 matches for 3 equilateral triangles.

b Use 8 matches for 3 equilateral triangles.

c Use 7 matches for 3 equilateral triangles.

d Use 12 matches for 6 equilateral triangles.

Note: A match can be used as a common side between two triangles.

10 Using 6 matches, construct four equilateral triangles. All triangles must have side

lengths of one match.

A Scalene

B Scalene and obtuse-angled

C Obtuse-angled and isosceles

D Isosceles and acute-angled

E Acute-angled and scalene

A Isosceles

B Right-angled

C Acute-angled and isosceles

D Isosceles and right-angled

E Right-angled and equilateral

A Right-angled

B Scalene and obtuse-angled

C Acute-angled and scalene

D Obtuse-angled and isosceles

E Right-angled and isosceles40°

40°

100°


History of mathematicsEUCLID (c . 300 BC)

During his life . . .

Alexander the Great dies.

Two of the Seven Ancient Wonders of the

World are built: the Colossus of Rhodes

and the Lighthouse of Alexandria.

Rome is supplied with fresh water by

aqueducts.

Very little is known about Euclid. He was a

Greek mathematician and was probably

educated at Plato’s Academy, in Athens.

Euclid developed an approach in which every

rule needed a proof. Using this method he

built up a set of theorems based upon

self-evident rules.

Euclid’s major work is a 13-volume

mathematical treatise spanning several topics

including the properties of numbers, plane

geometry and solid geometry. Elements is one

of the best known mathematical books ever

written. It contains 465 different mathematical

theories and proofs and has been in use

for more than fifteen hundred years.

Elements was first

printed in 1482 and

was still used as a

school textbook less

than a hundred

years ago. Even

today it provides a

basis for what is

taught in high

schools. Euclid’s

proofs included

most of the

geometrical facts

that we know today.

It is not known how

much of Elements is

Euclid’s original

work and how much he collected from other

sources. For example, Elements includes

Pythagoras’ theorem of right-angled triangles.

Euclid worked on finding the various types of

regular polygons and 3-dimensional

polyhedra. Some of the polygons he

researched were triangles (equilateral),

squares and nonagons (9-sided polygons), and

polyhedra such as tetrahedrons (4 faces),

dodecahedrons (12 faces) and icosahedrons

(20 faces).

In 306 BC, as King of Egypt, Ptolemy

founded a university at Alexandria where

Euclid became the first professor of

Mathematics. King Ptolemy wanted to know

if there was an easy way to learn geometry,

but Euclid said there was no ‘Royal road’ to

geometry and the King would have to learn it

by hard work!

Questions

1. What is the name of Euclid’s major

work?

2. When was this work first published

for general use?

3. Which king wanted to find an easy

method of learning geometry?


Angles in a triangleThe ancient Greek mathematician Euclid discovered that the angles in a triangle always

add up to 180°. This is true for any triangle no matter how large or small, wide or

narrow. We can use this law to find missing angles in triangles.

Find the value of the pronumeral in the following triangle.

THINK WRITE

The sum of angles in any triangle is

equal to 180°. Form an equation by

putting the sum of the angles on one side

and 180° on the other side of the equals

sign.

b + 40° + 63° = 180°

Solve for b; that is, subtract 103° from

both sides of the equation.

b + 103° = 180°

b + 103° − 103° = 180° − 103°

b = 77°

1

2

3WORKEDExample

63° 40°

b


THINK WRITE

The markings on the diagram indicate

that triangle ABC is isosceles with

AB = BC. The two base angles of an

isosceles are equal in size,

so ∠BCA = ∠BAC = 65°.

∠BCA = 65°

Form an equation by making the sum of

the angles in the triangle ABC equal to

180°.

h + 65° + 65° = 180°

Solve for h; that is, subtract 130° from


h + 130° = 180°

h + 130° − 130° = 180° − 130°

h = 50°

1

2

3

4WORKEDExample

65°

h

A

B

C



THINK WRITE

Form an equation by making the sum of

the angles in the given triangle equal to

180°.

w + w + 50° = 180°

Solve the equation to find the value of

w. First simplify by collecting like

terms.

2w + 50° = 180°

Subtract 50° from both sides of the

equation.

2w + 50° − 50° = 180° − 50°

2w = 130°

Divide both sides of the equation by 2. =

w = 65°

1

2

3

4 2w

2-------

130°

2-----------

5WORKEDExample

50°

w w


THINK WRITE

The angles in a triangle add up to 180°,

so form an appropriate equation.

x + 2x + 60° = 180°

Solve for x. First simplify by collecting

like terms.

3x + 60° = 180°


equation.

3x + 60° − 60° = 180° − 60°

3x = 120°

Divide both sides of the equation by 3. =

x = 40°

1

2

3

4 3x

3------

120°

3-----------

6WORKEDExample

60°x

2x

1. The sum of the angles in any triangle is 180°.

2. The base angles of an isosceles triangle are equal.

remember


Angles in a triangle

1 Find the value of the pronumeral in each of the following triangles.

a b

c d

e f


a b c

d e f


a b c

d e f

8B

Cabri Geometry

Anglesin right-angled

triangles

SkillSHEET

8.3

Anglesin a

triangle

WORKED

Example

3

Mathcad

Anglesin a

triangle

Cabri Geometry

Angle sumof a

triangle

60° 40°

t

55°

100 °

w

30°

m

30°50°

c

30°

60°

g80°

70°p

WORKED

Example

4 62°

q55°

x

50° t

45°

m k

k k p

WORKED

Example

5

34°

t

72°

s 96°

m

t48°

r

q

55°

t

x



a b c

d e f

5

The values for the pronumeral angles in the following diagrams are:

a A 180° B 117° C 27°

D 153° E 63°

b A 49° B 98° C 82°

D 44° E 180°

c A 90° B 30° C 60°

D 180° E 120°

6 In snowy regions, houses are built with steep roofs so the snow slides off. If the

sloping edges of the roof make an angle of 55° with its base, find the angle at the

apex.

7 A Christmas tree is shaped like an isosceles triangle. If the angle at the top is 36°, find

the size of each of the base angles.

8 A vertical flagpole is held up by a wire. The angle between the ground and the wire is

twice the angle between the pole and the wire. Find the angle between the pole and

the wire.

9 Maya measures the angles in a triangle, and finds that two angles are the same and

one is 15 degrees larger than the other two. What is the size of each angle?

10 The second-largest angle in a triangle is 5 degrees smaller than the largest angle and

5 degrees larger than the smallest angle. How large is each angle?

WORKED

Example

6 x

x x

60°2x

3x

4x

5x

2x + 1

x

131°2x – 1

3x + 6

45° 2x – 7

4x + 3

2x

multiple choice

63°

m

82°

t

2x

3x x


Exterior angles of a triangleIn exercise 8B you were dealing with the angles inside a triangle, called interior

angles. In this section you will look at the angles outside a triangle, called exterior

angles.

If one side of a triangle is extended, the angle between this

extension and the triangle is called an exterior angle.

Open the Cabri

Geometry file

‘Exterior angles of a

triangle’ on the

Maths Quest 8

CD-ROM, and

observe what happens

to the size of the

exterior and interior

angles when you

change the triangle.

Write your

conclusions. Can you

prove why this

relationship occurs?

e

e = exterior angle

Cabri Geometry

Exterioranglesof a

triangle

COMMUNICATION Exterior angles of a triangle

For the triangle shown, find the value of:

a x

b the exterior angle, e.

THINK WRITE

a Form an equation by making the

sum of angles in the given triangle

equal to 180°.

a x + 50° + 54° = 180°

Solve for x; that is, subtract 104°

from both sides of the equation.

x + 104° = 180°

x + 104° – 104° = 180° − 104°

x = 76°Continued over page

1

2

7WORKEDExample

50°

54°

x e


From the above example an important observation can be made:

An exterior angle and the interior angle adjacent to it are supplementary.

From worked example 7 we can also observe that the exterior angle, e, was equal to the

sum of the two interior angles not adjacent to it:

e = 104°

= 50° + 54°

In general,

In any triangle the exterior angle is equal to the sum of the two opposite interior

angles.

a + b = e

The following worked examples demonstrate the use of this rule.

THINK WRITE

b Angle x, together with exterior angle

e, makes a straight angle of 180°.

Write this as an equation.

b x + e = 180°

Substitute the value of x, found in

part a, into the equation.

76° + e = 180°

Solve for e; that is, subtract 76° from


76° – 76° + e = 180° − 76°

e = 104°

1

2

3

ea

b

Find the size of the exterior angle in the triangle shown.

THINK WRITE

The exterior angle equals the sum of

the two opposite interior angles.

State this as an equation.

e = 30° + 80°

Simplify to find the value of e. e = 110°

1

2

8WORKEDExample

30°

80°

e


Find the value of the pronumeral in the triangle shown.

THINK WRITE

The exterior angle of a triangle equals

the sum of the two opposite interior

angles. Form an appropriate equation.

w + 70° = 132°


equation to find the value of w.

w + 70° – 70° = 132° – 70°

w = 132° − 70°

= 62°

1

2

9WORKEDExample

132°

70°

w

Find the value of the pronumeral in the triangle shown.

THINK WRITE

The exterior angle of a triangle equals

the sum of the two opposite interior

angles. Form an appropriate equation.

4x + 2x = 120°

Solve for x. First collect like terms. 6x = 120°

Divide both sides of the equation by 6

to find the value of x.

=

= 20°

1

2

36x

6------

120

6---------

10WORKEDExample

120°4x

2x

1. When one side of a triangle is extended, the angle between this extension and

the triangle is called an exterior angle of the triangle.

2. The exterior angle and the interior angle adjacent to it are supplementary (add

up to 180°).

3. In any triangle, an exterior angle is equal to the sum of the two opposite

interior angles.

remember


Exterior angles of a triangle

1 For each of the triangles shown, find the value of:

i angle x

ii the exterior angle, e.

a b c

2 a Copy and complete this table, showing the interior angles and the opposite

exterior angle for the triangles in question 1.

b Verify the rule connecting the interior angles and the opposite exterior angle for

each triangle.

3 Find the size of the exterior angle in these triangles.

a b c d


a b c

5 Find the value of the pronumeral(s) in each triangle shown.

a b c

d e f

QuestionGiven interior

anglesSum of given

interior anglesOpposite exterior

angle, e

1a 20°, 60° 80°

1b

1c

8C

SkillSH

EET 8.4

Complementary angles

SkillSH

EET 8.5

Supplementary angles

Mat

hcad

Exteriorangles of a triangle

WORKED

Example

7

ex60°

20°

ex42°

65°

e x 60°

60°

WORKED

Example

8

t30°

72° n

25° 65°

m

54°

q83°

32°

WORKED

Example

9

t

110°

65°w

90°43° m

132°

d

k

114°

WORKED

Example

10

2x

3x

120°

x

x + 4

60°

3x

5x50°

2x

6x

y

x

x + 30

70° y

x

2x

60°


6

a The value of the pronumeral, e, in this triangle is:

b The value of the pronumeral, w, in this triangle is:

c The value of the pronumeral, k, in this triangle is:

d The value of the pronumeral, m, in this triangle is:

7 Find the size of the exterior angle of an equilateral triangle.

8 The roof of an A-frame house makes a 65° angle with the

ground. What angle does the roof make with the balcony?

A 48° B 69° C 117° D 63° E 180°

A 98° B 54° C 44° D 82° E 180°

A 59° B 62° C 118° D 180° E 121°

A 42° B 126° C 63° D 84° E 56°

multiple choice

e

69°

48°

w 98°

54°

k 59°

m

2m

126°

b

65°

Roof

Balcony

Ground


9 In this Native American teepee, the poles meet at an

angle of 50°.

a Find the size of the acute angle that each pole

makes with the ground.

b Find the size of the obtuse angle that each pole

makes with the ground.

10 The two sloping sides of a roof meet at an apex

angle of 102°. Find the obtuse angle that each of

the sloping sides makes with the horizontal.

11 The three exterior angles of a triangle are 105°, 125° and 130°.

Find the interior angles.

QuadrilateralsA quadrilateral is any closed 2-dimensional shape with four straight sides. All

quadrilaterals can be subdivided into two major groups: parallelograms and other

quadrilaterals. Parallelograms are quadrilaterals with two pairs of opposite sides being

parallel.

Note: Parallel lines are those lines that never meet. We indicate that the lines are

parallel by placing identical arrows on each line.

1 Find the size of each of the exterior angles in the four triangles shown.

2 Find the sum of the exterior angles in each of the above triangles.

3 Is there any pattern that you have observed? Complete this sentence: In any

triangle the sum of the exterior angles is .

a b

50°50°

130°

105°

125°

GAM

E time

Geometry— 001

THINKING Exterior angles

30°

80°

65°

110°35°

50°


The table below shows quadrilaterals, which belong to either of these two groups,

and their properties.

Parallelograms Opposite sides are parallel

Square All sides are equal in length.All angles are 90°.

Rectangle Opposite sides are equal in length.All angles are 90°.

Rhombus All sides are equal in length.Opposite angles are equal in size.

Parallelogram Opposite sides are equal in length.Opposite angles are equal in size.

Other quadrilaterals Opposite sides not all parallel

Trapezium One pair of parallel sides.

Kite Two pairs of adjacent (next to each other) sides are equal in length.Angles between unequal sides are equal in size.

Irregular quadrilateral

Looks like none of the above (possesses no special properties).

Cabri Geometry

Squares

Cabri Geometry

Rectangles

Cabri Geometry

Rhombuses

Cabri Geometry

Parallelograms

Cabri Geometry

Trapeziums

Cabri Geometry

Kites


Quadrilaterals

1 Copy and complete this table. Allow enough space to fit in the missing figures.

(continued)

Picture Name Definition

Parallelogram

All sides are of equal length; opposite angles are equal in size.

State whether each of the following is true or false.

a The opposite sides of any rectangle are parallel.

b Any rectangle is a square.

THINK WRITE

a All rectangles are parallelograms. By definition, a

parallelogram has two pairs of opposite sides parallel.

Since a rectangle is a parallelogram, it must have a

parallelogram’s properties. Therefore, opposite sides of

any rectangle are parallel and the statement is true.

a True

b By definition, both a square and a rectangle must have

four right angles. However, in a square all four sides are

the same, but in a rectangle only opposite sides must be

of equal length. Therefore, the statement is false.

b False

11WORKEDExample

1. A quadrilateral is any closed 2-dimensional shape with four straight sides.

2. All quadrilaterals can be subdivided into two major groups: parallelograms

(these include rectangles, squares, parallelograms and rhombuses) and other

quadrilaterals (these include trapeziums, kites and irregular quadrilaterals).

3. Parallelograms are quadrilaterals with two pairs of opposite sides parallel.

remember

8D

Cabri

Geometry

Types of quadrilaterals

Mat

hcad

Classifyingquadrilaterals


2 State whether each of the following is true or false.

a All squares are rectangles.

b All squares are rhombuses.

c A trapezium could have two sides of equal length.

d A parallelogram with adjacent sides being equal is a square.

e A parallelogram with at least one right angle is a rectangle.

f A kite could have one right angle.

g A kite cannot have two right angles.

h An irregular quadrilateral cannot contain a 90° angle.

i A rectangle is a quadrilateral because it has two pairs of parallel sides.

j A rhombus is a parallelogram because it has two pairs of parallel sides.

k Not every parallelogram has opposite sides equal in length.

l Not every square is a parallelogram.

3 For each of the following, state the name of the quadrilateral that best matches the

clues.

a I am a parallelogram with all sides of equal length. What am I?

b I have two pairs of equal sides. These sides are not opposite. What am I?

c I am a rhombus with 90-degree angles. What am I?

4 Construct a trapezium that has:

a two adjacent right angles

b two sides (that are not parallel) of equal length.

5 Draw the solutions to each of the following problems in your workbook.

a With one line, divide a square into two rectangles.

b With one line, divide a square into two trapeziums.

c With one line, divide a rectangle into a square and another

rectangle.

d With one line, divide a rhombus into two parallelograms.

e With one line, divide a rhombus into two trapeziums.

f With four lines, divide a square into seven squares.

Picture Name Definition

Kite

Irregular quadrilateral

All sides equal; all angles are 90°.

WORKED

Example

11


6

The walls of the metal shed in the foreground

of the photograph at right are:

A squares and parallelograms

B irregular quadrilaterals and squares

C rhombuses and rectangles

D rectangles and trapeziums

E rectangles and squares

1 Copy the kite at right onto a piece of paper. Cut it out

and then cut along the dotted lines to form four triangles.

Rearrange these four triangles to form:

a a rectangle

b a trapezium

c a parallelogram.

Hint: You may flip pieces upside down.

2 Use two sticks or cut drinking straws of length 3 cm and two of length 5 cm.

How many different types of quadrilateral can you make? List them and draw

the solutions in your workbook.

WorkS

HEET 8.1

multiple choice

DESIGN Forming quadrilaterals

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 How many rectangles can be found

in this figure?

2 How would you cut this trapezium into four

pieces of exactly the same size and shape?

The trapezium is made up of a square and

half of a similar square divided diagonally.


1 Using side and angle markings where appropriate, draw a scalene triangle.

2 Using side and angle markings where appropriate, draw an acute-angled isosceles

triangle.

3 Find the value of the pronumeral in the diagram below.

4 Find the value of the pronumeral in the diagram below.

5 Beth was drawing a 2-dimensional diagram of a pear. She noticed that it resembled an

isosceles triangle. If the angle at the top is 42°, find the base angle.

6 True or false? The interior angle marked x in the triangle shown is x = 90°.

7 The three interior angles of the triangle are 54°, 61° and 65°. Find the exterior angles

a, b and c.

8 Joshua placed 2 playing cards on an angle to form the sides of

a card house. The cards met at an angle of 80°. Find the obtuse

angle that each card makes with the surface of the table.

9 Draw and name a shape with 4 sides where the opposite sides are equal in length and

opposite angles are equal. (There are 2 pairs of parallel sides.)

10

A quadrilateral has two 5-cm-long sides and two 10-cm-long sides. If all the angles

are equal, this is best described as a:

A rhombus B square C rectangle D trapezium E parallelogram

1

112°

10°

x

40°

2x +3

4x –7

10° x

100°

61°

54°

65°

c

a

b

80°

a b

multiple choice


Angles in a quadrilateralYou probably already know the following fact.

The sum of the interior angles in any quadrilateral is equal to 360°.

Check that this works by investigating the angles in a quadrilateral using the Cabri

Geometry file supplied on the Maths Quest 8 CD-ROM.

This rule applies to any quadrilateral, regardless of shape or size. If you walk around

a quadrilateral, you will end up at your starting point.

The rule can be used to find missing angles in quadrilaterals, as shown in the worked

examples that follow.

Cabri

Geometry

Angles in a quadrilateral

Find the value of the pronumeral in the diagram at right.

THINK WRITE

The sum of angles in the quadrilateral

must be 360°, so form an appropriate

equation. (The two angles marked with

the small square are each 90°.)

b + 110° + 90° + 90° = 360°

Solve for b; that is, subtract 290° from


b + 290° = 360°

b + 290° – 290° = 360° − 290°

b = 70°

1

2

12WORKEDExample

b

110°

Find the value of the pronumeral in the diagram at right.

THINK WRITE

Form an equation by putting the sum of

angles on one side and 360° on the

other side of the equals sign.

t + t + 75° + 85° = 360°

Simplify by adding numbers together

and collecting like terms.

2t + 160° = 360°


equation.

2t + 160° – 160° = 360° − 160°

2t = 200°

Divide both sides of the equation by 2

to find the value of t.

=

t = 100°

1

2

3

42t

2-----

200°

2-----------

13WORKEDExample

t t

85°

75°


Angles in a quadrilateral

1 Find the value of the pronumeral in each of the following diagrams.

a b c

d e f

Find the value of x in the quadrilateral shown below.

THINK WRITE

Add the four angles and set them equal

to 360°.

x + 2x + 3x + 10° + 110° = 360°

To solve the equation, first simplify by

collecting like terms.

6x + 120° = 360°


equation.

6x + 120° – 120° = 360° − 120°

6x = 240°

To find the value of x, divide both sides

of the equation by 6.

=

x = 40°

1

2

3

46x

6------

240°

6-----------

14WORKEDExample

x

2x

3x + 10 110°

The sum of the interior angles in any quadrilateral is equal to 360°.

remember

8ESkillSHEET

8.6

Angle sumof a

quadrilateral

Mathcad

Angles in aquadrilateral

WORKED

Example

12

Cabri Geometry

Angle sumof a

quadrilateral

Cabri Geometry

Angles in aquadrilateral

x95°

85°

50°

x

45° 50°

100°t

112°

112°68°

m

95° 95°

50°

k

60°

95°120°

q


2

a In this square, the equal angles are:

b In this kite, the pairs of equal angles are:

c In this parallelogram, the pairs of equal angles are:

d In this trapezium, the pairs of equal angles are:

3 Find the value of the pronumeral(s) in each of the following diagrams.

a b c

d e f

A a and b only

B a and c only

C a and b are equal, c and d are equal

D all angles

E a and c are equal, b and d are equal

A a and c only

B a and c are equal, b and c are equal

C a and c are equal, b and d are equal

D b and d only

E all angles are equal

A a and c only

B a and d are equal, b and c are equal


D b and d only


A b and c only

B a and d are equal, b and c are equal


D a and d only


multiple choice

a b

d c

c

d b

a

a b

d c

a

d c

b

WORKED

Example

13 m

m

74°

74°t t

98°

42°

k k

115° 115°

85°

122° q

s

105°t

m m 116°p

q r


4 Find the value of x in each of the following quadrilaterals.

a b c

d e f

g h

i

5 The swing at the local

park is in the shape of a

trapezium. If the angle

that the pole makes with

the ground is 70°, then

find the angle, x,

between the pole and the

top crossbar.

6 Jennifer built a kite. If the angle at the top of the kite is three times

more than the angle at the tail, and the angles on the side are 80°, find

the angle at the tail.

7 One angle in a parallelogram is double the other angle. Find all

angles.

8 One angle in a rhombus is 40°. Find all other angles.

9 A certain quadrilateral has each angle 10° greater than the previous one, except the

smallest angle. How large is the smallest angle?

WORKED

Example

14

x

2x

x

3x

2x

2xx – 10

x + 10 110°

4x

4x

2x

2x 2x + 6

x + 8

98°

4x + 30

2x + 40

2x – 20

2x + 40

x + 22

x + 282x – 15

x 2x + 9

x – 22 2x – 32

3x – 27

4x – 3

3x + 3

3x – 12

2x + 12

80° 80°

3x

x

What did the pencil say to the eraser?

64o

75o 34o 27o 108o 252o 108o 75o 134o

45o 134o 124o 96o 96o 124o 12o 108o 96o

108o 108o 64o 252o 38o 124o 124o 38o 18o 124o 54o

The size of the lettered anglesgives the puzzle answer code.

57o D

52o

E

98o

71o

83o59o

A

72oR46o

N

126o

105o

OS

63oJ

45o X

39o41o

28o

LK 56o

112o

37o

Q

47o

56o

Z

56o

T



Angles in polygonsIn the previous exercises you saw that the sum

of the angles in a triangle is 180° and the sum

of the angles in a quadrilateral is 360°.

In this section you will learn a method of finding the sum of the angles in any polygon.

1 For every polygon shown in the table below, draw as many diagonals, as

possible from the vertex, marked X. (This will divide each polygon into a

number of triangles.)

2 Copy and complete this table:

3 Can you see a pattern? What would be the sum of the angles in a dodecagon (12

sides)? Can you predict the angle sum of an icosagon (20 sides)? What about a

polygon with 100 sides?

a b

c

a + b + c = 180°

a b

cd

a + b + c + d = 360°

THINKING Sum of angles in a polygon

Polygon NameNumber of

sidesNumber of triangles

Sum of angles

Triangle 3 1 180o

Quadrilateral 4 2 2 × 180°

= 360o

Pentagon 5 3 3 × 180o

= 540o

Hexagon 6 4

Heptagon 7

Octagon 8

Decagon 10

X

X

X

X

X

X

X


The patterns that you have observed in the previous investigation can be generalised as

follows: For any polygon the sum of interior angles = 180° × (number of triangles).

Furthermore, the number of triangles = the number of sides − 2.

Therefore,

The sum of the interior angles in any polygon = 180° × (n − 2 ), where n is the

number of sides of the polygon.

We can use this formula for finding the size of unknown angles in various polygons,

as shown in the following worked examples.

Find the sum of the interior angles of the polygon shown.

THINK WRITE

Write the general formula for the angle

sum of a polygon.

Sum of angles = 180° × (n − 2)

Count the number of sides in the given

polygon to identify the value of n.

n = 11

Substitute the value of n into the

formula and evaluate.

Sum of angles = 180° × (11 − 2)

= 180° × 9

= 1620°

1

2

3

15WORKEDExample

For the polygon shown at right find:

a the sum of its interior angles

b the value of the pronumeral.

THINK WRITE

a Write the general formula for the

angle sum of a polygon.

a Sum of angles = 180° × (n − 2)

The shape has five sides, so state the

value of n.

n = 5

Substitute the value of n into the

formula and evaluate.

Sum of angles = 180° × (5 − 2)

= 180° × 3

= 540°

b Form an equation by making the

sum of interior angles equal to 540°.

b p + 120° + 130° + 50° + 70° = 540°

Solve for p; that is, subtract 370°


p + 370° = 540°

p + 370° – 370° = 540° − 370°

= 170°

1

2

3

1

2

16WORKEDExample120°

70°50°

130°

p


Regular polygons

A regular polygon is the one in which all sides are equal in length and all angles

are equal in size.

If a polygon is regular, we can find the size of each of its angles by first finding the sum

of the interior angles of the polygon and then dividing it by the number of angles.

Consider the following example.

1 What is the sum of the interior

angles of a stop sign?

2 What is the size of each angle?

3 What is the size of each exterior

angle?

4 What is the angle sum and size of

each angle for a give-way sign?

Find the value of the pronumeral in this regular polygon.

THINK WRITE

Write the general formula for the sum of angles

in a polygon.

Sum of angles = 180° × (n − 2)

The shape has five sides, so state the value of n. n = 5

Substitute the value of n into the formula and

evaluate.

Since the polygon is regular, all of its angles are

equal in size. So, to find the size of each angle,

divide the sum of angles by the number of

angles.

Note: The number of angles corresponds to the

number of sides, n.

Sum of angles = 180° × (5 − 2)

= 180° × 3

= 540°

a = 540° ÷ 5

= 108°

1

2

3

4

17WORKEDExample

a

1. The sum of the interior angles in any polygon equals 180° × (n − 2), where n is

the number of sides of the polygon.

2. A regular polygon has all sides equal in length and all angles equal in size.

3. To find the size of the angles in a regular polygon, find the sum of its angles

first and then divide it by the number of angles in the polygon.

remember

THINKING Angles in a regular polygon


Angles in polygons

1 Find the sum of the interior angles of each of the polygons shown.

a b c

2 For each of the polygons shown, find:

i the sum of its interior angles

ii the value of the pronumeral.

a b c

d e f

3 State whether each of the following polygons is regular, or not. Give reasons.

a b c

d e f

8F

SkillSH

EET 8.7

Angle sumof apolygon

Mat

hcad

Angles in polygons

Cabri

Geometry

Star polygons

Cabri

Geometry

Exterior angles of a polygon

Cabri

Geometry

Anglesum of a polygon

WORKED

Example

15

WORKED

Example

16

120° 120°

b

150° 150°

110°c

55°

260°

240°55°

170°

d

45°

250°

150° 150°

45°

h

6x – 10

3x + 11

8x + 10

3x – 6

5x

4x – 14

3x + 3

14x + 2

15x – 25

5x – 9

6x + 6

6x – 25

7x + 2


4 Find the value of the pronumeral in these regular polygons.

a b c d

5 Some regular shapes have special names. Draw each of these shapes, marking all

equal sides and angles, and write down their common name:

a a regular quadrilateral b a regular triangle.

6 Two angles of a pentagon are right angles. The other three angles are all equal. Find

the size of these angles.

7 Two angles of a hexagon are right angles. The other four angles are all equal. Find the

size of these angles.

8 A cross as shown in the diagram at right is a polygon.

a What is the name of this polygon?

b What is the sum of its angles?

9 Yvette draws a regular 15-sided polygon. How large is each angle?

10 Sam draws a regular 30-sided polygon. How large is each angle?

Angles and parallel linesIn previous exercises we have discussed angles in triangles,

quadrilaterals and other polygons. In this section we will

investigate different angles associated with parallel lines.

A line, intersecting a pair (or a set) of parallel lines, is

called a transversal.

Cutting parallel lines by a transversal creates a number of angles. These angles are

related in a number of ways, as we will now see.

The diagram at right shows a regular triangle,

quadrilateral, pentagon and hexagon, constructed on a

common base 2 cm long. Using a ruler and a

protractor, construct, on a common base 5 cm long,

each of the following polygons:

a a regular triangle

b a regular quadrilateral

c a regular pentagon

d a regular hexagon

e a regular octagon

f a regular decagon

g a regular dodecagon.

Hint: Calculate the size of each angle first.

WORKED

Example

17

tp

a

t

Cabri Geometry

Regularpolygons

COMMUNICATION Regular polygons

2 cm

Transversal

Parallel lines


Vertically opposite anglesThe diagram at right shows two vertically opposite angles.

Vertically opposite angles are equal in size.

Thus, in the diagram at right ∠a = ∠b.

Vertically opposite angles are often associated with an

X shape.

Corresponding anglesThe diagram at right shows two angles, a and b, positioned

below the parallel lines to the right of a transversal.

When both angles are on the same side of the transversal

(both to the left or both to the right of it) and are either both

above or both below the parallel lines, such angles are called

corresponding angles.

Corresponding angles are equal in size.

Thus, in the above diagram ∠a = ∠b.

The position of corresponding angles is easy to remember

by thinking of it as F-shaped.

Co-interior (or allied) anglesThe diagram at right shows two angles, a and b, positioned

‘inside’ the parallel lines, on the same side (to the right) of

the transversal. Such angles are called co-interior or allied

angles.

Co-interior angles are supplementary; that is, they add up

to 180°.

Thus, in the diagram at right ∠a + ∠b = 180°.

The position of the co-interior angles is easy to remember

by thinking of it as C-shaped.

Alternate anglesThe diagram at right shows two angles, a and b, positioned

‘inside’ the parallel lines on alternate sides of the transversal.

Such angles are called alternate angles.

Alternate angles are equal in size

Thus, in the above diagram ∠a = ∠b.

The position of alternate angles is easy to remember by

thinking of it as Z-shaped.

Cabri

Geometry

Verticallyoppositeangles

a b

a

b

Cabri

Geometry

Correspondingangles

a

b

a

b

Cabri

Geometry

Co-interiorangles

a

b

a

b

Cabri

Geometry

Alternateangles

a

b

a

b


Calculating angles associated with parallel linesAngle relationships associated with parallel lines can be used to find the size of missing

angles, as shown in the following worked examples.

Open the Cabri Geometry file ‘Parallel lines’ on the Maths Quest 8 CD-ROM.

Follow the instructions on the screen to vary the position of the transversal and

observe the change in the angles. Identify the different angle relationships.

Cabri Geometry

Parallel

lines

THINKING Angle relationships with parallel lines

For the diagram at right:

a state the type of angle relationship

b find the value of the pronumeral.

THINK WRITE

a Study the diagram: which shape —

X, Z, F or C — would include both

angles that are shown? Copy the

diagram into your workbook and

highlight the appropriate shape.

a

State the name of the angles

suggested by the C shape.

Shown angles are co-interior.

b Co-interior angles add to 180°.

Write this as an equation.

b m + 100° = 180°

Solve for m; that is, subtract 100°


m + 100° – 100° = 180° − 100°

m = 80°

1

m

100º

2

1

2

18WORKEDExample

m

100º

Find the value of the pronumerals in the diagram shown at

right. Provide reasons for your answers.

Continued over page

THINK WRITE

Identify the relationship between angle

a and 58° and form an equation.

a + 58° = 180° (straight line)

Solve for a; that is, subtract 58° from


a + 58° − 58° = 180° − 58°

a = 122°

1

2

19WORKEDExample

77º65º

a 58º d fe

b c g


THINK WRITE


b and 58° and state the value of b.

b = 58° (alternate angles)


b, 77°, and angle c, and then form an

equation.

b + 77° + c = 180° (straight line)

58° + 77° + c = 180°

Solve for c; that is, subtract 135° from


135° − 135° + c = 180° − 135°

c = 45°

Identify the relationship between 58°,

77°, and angle d, and then form an

equation.

58° + 77° + d = 180° (angle sum of

58° + 77° + d = 180° triangle)

Solve for d; that is, subtract 135° from


Note: Angles c and d are alternate and

therefore equal.

135° − 135° + d = 180° − 135°

d = 45°


g and 65° and form an equation.

g + 65° = 180° (straight line)

Solve for g; that is, subtract 65° from


g + 65° − 65° = 180° − 65°

g = 115°

Identify the relationship between angles

f and g and form an equation.

g + f = 180° (co-interior angles)

115° + f = 180°

Solve for f; that is, subtract 115° from


Note: Angle f and 65° are alternate and

therefore equal.

115° − 115° + f = 180° − 115°

f = 65°

Identify the relationship between angles

d, e and f, and then form an equation.

d + e + f = 180° (straight line)

45° + e + 65° = 180°

110° + e = 180°

Solve for e; that is, subtract 110° from


110° − 110° + e = 180° − 110°

e = 70°

3

4

5

6

7

8

9

10

11

12

13

1. Vertically opposite (X) angles are equal in size.

2. Corresponding (F) angles are equal in size.

3. Co-interior (C) angles add to 180°.

4. Alternate (Z) angles are equal in size.

5. Supplementary angles add to 180°.

6. Complementary angles add to 90°.

7. If the given angles are in none of the above relations, we might need to

find some other angle first. This other angle must be related to both given

angles.

remember


Angles and parallel lines

1 a Copy the diagram into your workbook. Clearly draw the F shape

on your diagram and label the angle corresponding to the one

that is marked.

b Copy the diagram into your workbook. Clearly draw the Z shape

on your diagram and label the angle alternate to the marked

angle.

c Copy the diagram and label the angle vertically opposite to the

marked angle. Clearly draw the X shape on your diagram.

d Copy the diagram and label the angle co-interior to the marked

angle. Clearly draw the C shape on your diagram.

2 Match each diagram with the appropriate name from the five options

listed below.

Diagram Name

a A Vertically opposite angles (X)

b B Co-interior angles (C)

c C Corresponding angles (F)

d D Alternate angles (Z)

8G

SkillSHEET

8.8

Anglesand

parallellines


e E None of the above

3

In the diagram at right,

a which angle is vertically opposite to angle p?

A j B k C m D r E q

b which angle is corresponding to angle p?

A j B k C m D r E q

c which angle is co-interior to angle p?

A j B k C m D r E q

d which angle is alternate to angle p?

A j B k C m D r E q

4 In the diagram at right, list all pairs of:

a vertically opposite angles

b corresponding angles

c co-interior angles

d alternate angles.

5 For each of the following diagrams:

i state the type of angle relationship

ii find the value of the pronumeral.

a b c

d e f

6 Find the value of the pronumerals in each of the following diagrams.

a b c

multiple choice

r

n

k

j q

m

p

t

ac

eg h

f

db

SkillSH

EET 8.9

Angle relationships

WORKED

Example

18

60°

p45°

q

65°

s

72°t 70°m

132°

n

64°116°

y

38°

62°

z44°

44°

b


d e f

7 Find the value of the pronumerals, stating the type of angle relationship, in each of the

following diagrams.

a b

c d

8 Find the value of the pronumerals in each of the following diagrams.

a b c

d e f

9 If the angle co-interior to x is 135°, find the size of angle x.

10 If the angle corresponding to y is 55°, find the size of angle y.

11 A hill is at an angle of 30° to the horizontal. A fence is put in,

consisting of a railing parallel to the ground and vertical fence

posts. Find the angle between the top of the fence post and the

rail.

68°

72° g135°

h

120° 110°

k

SkillSHEET

8.10

Moreangle

relationships

WORKED

Example

19

59º

48ºp q

rs

73º

42ºzy

x

w

42ºe

b

a126º

c

d

47º

64º

ce

ba

d

b

123°

x

137°

y

62°

zz

80°

160°

p

qq

30°

p


12 Two gates consist of vertical posts, horizontal struts and diagonal beams. Find the

angle, a, as shown in the gates below.

13 Two transversal lines cross a pair of parallel lines at 120°

and 135°. Find the angle between the transversals.

14 Find the angles w, x, y, z in the diagram shown.

40°

aa b

GAM

E time

Geometry— 002

WorkS

HEET 8.2

t

120° 135°

x yz

w

45°

60°

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 Is the line AB parallel to line CD?

Explain your answer.

2 This figure has 8 unit triangles. Remove 4

matches so that 4 unit triangles are left

behind.

3 Move 4 matches so that there are 4 unit

squares instead of 5.

C

48°

84°

133°

A

D

B


1 Using side and angle markings where appropriate, draw a right-angled scalene

triangle.

2 a True or false? The name of the quadrilateral shown is

a trapezium.

b True or false? The complement of 25° is 75°.

3 Name the following pair of angles in this diagram.

For questions 4 to 9, find the value of the pronumerals.

4

5 6

7 8 9

10 Find the angles v, w, x, y, z in the diagram shown.

2

70° x

53°x

y

172°y

120°

15°

35°

ym

140º 140º 80°

u

v

z y

w

x

58º

37º

v

w x y z


Constructing trianglesUsing a ruler, protractor and a pair of compasses, you can

construct any triangle if you are given three side lengths,

two side lengths and the angle between them, or two

angles and the length of the side between them.

Constructing a triangle given three side lengthsIf the lengths of the three sides of a triangle are known,

it can be constructed with the help of a ruler and a pair

of compasses, as shown in the following worked

example.

Constructing a triangle given two angles and the side between themIf the size of any two angles of a triangle and the length of the side between these two

angles are known, the triangle can be constructed with the aid of a ruler and a

protractor. The following worked example shows how this is done.

Using a ruler and a pair of compasses, construct a triangle with side lengths 15 mm,

20 mm and 21 mm.

THINK DRAW

Rule a line representing the longest side

(21 mm).

Open the pair of compasses to the

shortest side length (15 mm).

Draw an arc from one end of the

21 mm side.

Open the pair of compasses to the

length of the third side (20 mm) and

draw an arc from the other end of the

21 mm side.

Join the point of intersection of the two

arcs and the end points of the 21 mm

side with lines. Erase the arcs.

121 mm

2

3

21 mm

15 mm

4

21 mm

15 mm

20 mm

5

21 mm

15 mm 20 mm

20WORKEDExample


Constructing a triangle given two sides and the angle between themA triangle can be constructed using a protractor and a ruler, if the lengths of two sides

and the size of the angle between them (called an included angle) are given. The

following worked example shows how this can be done.

Use a ruler and protractor to construct a triangle with angles 40° and 65° and the side

between them of length 2 cm.

THINK DRAW

Rule a line of length 2 cm.

Place the centre of the protractor on one

endpoint of the line and measure out a

40° angle. Draw a line so that it makes

an angle of 40° with the 2 cm line.

Place the centre of the protractor on the

other endpoint of the 2 cm line and

measure an angle of 65°. Draw a line so

that it makes a 65° angle with the 2 cm line.

If necessary, continue the lines until they

intersect each other to form a triangle.

Erase any extra length.

1

2

2 cm

40°

3

2 cm

40° 65°

4

2 cm

40° 65°

21WORKEDExample

Use a ruler and protractor to construct a triangle with sides 6 cm and 10 cm long and an

angle between them of 60°.

THINK DRAW

Rule a line of length 10 cm.

Place the centre of the protractor on one

endpoint of the line and mark an angle

of 60°.

Note: These figures have been reduced.

Join the 60° mark and the endpoint of

the 10 cm side with the straight line.

Extend the line until it is 6 cm long.

Join the endpoints of the two lines to

complete the triangle.

1

2

10 cm

60°

3

10 cm

6 cm

60°

4

10 cm

6 cm

60°

22WORKEDExample


Constructing triangles

1 Using a ruler and pair of compasses, construct triangles with the following side lengths:

a 7 cm, 6 cm, 4 cm b 5 cm, 4 cm, 5 cm

c 6 cm, 5 cm, 3 cm d 6 cm, 6 cm, 6 cm

e 7.5 cm, 4.5 cm, 6 cm f 2 cm, 6.5 cm, 5 cm

g an equilateral triangle of side 3 cm h an equilateral triangle of side 4.5 cm.

2 Use a ruler and protractor to construct these triangles:

a angles 60° and 60° with the side between them 5 cm long

b angles 50° and 50° with the side between them 6 cm long

c angles 30° and 40° with the side between them 4 cm long

d angles 60° and 45° with the side between them 3 cm long

e angles 30° and 60° with the side between them 4 cm long

f angles 65° and 60° with the side between them 3.5 cm long

g angles 60° and 90° with the side between them 5 cm long

h angles 60° and 36° with the side between them 4.5 cm long.

3 Use a ruler and protractor to construct the following triangles:

a two sides 10 cm and 5 cm long, angle of 30° between them

b two sides 8 cm and 3 cm long, angle of 45° between them

c two sides 6 cm and 6 cm long, angle of 60° between them

d two sides 4 cm and 5 cm long, angle of 90° between them

e two sides 7 cm and 6 cm long, angle of 80° between them

f two sides 9 cm and 3 cm long, angle of 110° between them

g two sides 6 cm and 6 cm long, angle of 50° between them

h two sides 5 cm and 4 cm long, angle of 120° between them.

4 a Use a ruler and a pair of compasses to draw an isosceles triangle with two sides

5 cm long and one side 7 cm.

b Use your protractor to measure the size of the largest angle.

c Complete this sentence using one of the words below: This triangle is an

-angled triangle.

i acute ii right iii obtuse

1. A triangle can be constructed using a ruler and a pair of compasses if the three

sides are known.

2. If two angles of a triangle and the side between them, or two sides and an angle

between them are known, the triangle can be constructed using a protractor and

a ruler.

3. When using a protractor:

(a) make sure that the baseline of the protractor is exactly on the line, and the

centre of the protractor’s baseline (that is, where the vertical (90°) line

intersects with the horizontal base line) is exactly on the point from which

you are measuring the angle.

(b) use the scale that begins from 0° (not 180°).

remember

8H

SkillSH

EET 8.11

Measuring and drawing lines

WORKED

Example

20

SkillSH

EET 8.12

Constructing angles with a protractor

Cabri

Geometry

Threesides

Cabri

Geometry

Two angles and a side

Cabri

Geometry

Two sides and an angle between

WORKED

Example

21

WORKED

Example

22


5 a Use a ruler and a pair of compasses to draw a scalene triangle with sides 8 cm,

10 cm and 13 cm.

b Use your protractor to measure the size of the largest angle.

c Complete this sentence using one of the words below: This triangle is a

-angled triangle.

i acute ii right iii obtuse

6 Imagine that you were to copy a given triangle, using any tool(s) from your set of a

ruler, a protractor and a pair of compasses. What is the minimum information you

would need to accomplish the task? (There is more than one possible answer.)

Isometric drawingWhen working with 3-dimensional

models and designs, it is often

useful to have the design or model

drawn on paper (that is, in 2 dimen-

sions).

An isometric drawing is a

2-dimensional drawing of a

3-dimensional object.

This picture shows an architect’s

drawing of a beach hut and environs

in isometric view superimposed on

the actual hut. Architects and drafts-

persons often use isometric

drawings to give their clients a

clear picture of the proposed design.

First copy the incomplete figure at far right onto

isometric dot paper. Complete the isometric

drawing of the object shown at near right.

THINK DRAW

Study the object and identify the lines that

have already been drawn. Fill in the

missing lines on your isometric drawing to

match the object.

23WORKEDExample


Isometric drawing

1 Copy the following figures onto isometric dot paper and complete the isometric

drawing of the objects shown.

a b

Draw the following object on isometric dot paper.

(You could construct it first from a set of cubes.)

THINK DRAW

Use cubes to make the object shown

(optional). Draw the front face of the

object. The vertical edges of the

3-dimensional object are shown with

vertical lines on the isometric drawing;

the horizontal edges are shown with the

lines at an angle (by following the dots

on the grid paper).

Draw the left face of the object.

Add the top face to complete the

isometric drawing of the object.

1

2

3

24WORKEDExample

1. An isometric drawing is a 2-dimensional drawing of a 3-dimensional object.

2. If possible, construct the solid from the set of cubes prior to drawing its

isometric view.

3. Draw the front face first.

4. In isometric drawings, vertical edges of a 3-dimensional object are shown with

vertical lines, while horizontal edges are shown with the lines drawn at an

angle.

remember

8IWORKED

Example

23


c d

2 Draw each of the following objects on isometric dot paper. (You might wish to make

them first from a set of cubes.)

a b

c d

3 Construct the following letters using cubes, and then draw the solids on isometric dot

paper:

a the letter T with 5 cubes b the letter L with 7 cubes

c the letter E with 10 cubes d the letter H with 7 cubes.

4 Draw these objects, whose front (F), right (R) and top (T) views are given, on isometric

dot paper.

a b

c d

5 Draw the front, right and top views of these objects.

a b c d

WORKED

Example

24

F R T

F R

T

F

RT F R T


6 Draw the following figure on isometric dot paper.

7 Draw a selection of buildings from this photograph of the Melbourne skyline on iso-

metric dot paper.


Geometric constructionsUsing your imagination, a sharp pencil and eraser, a ruler, protractor and a pair of

compasses you can create some interesting geometric designs.

Use the following steps to construct a pentagonal star.

Step 1. Draw a circle of radius 1 cm.

Step 2. Draw a pentagon in the circle. (Mark off every 72°.)

Step 3. Join all vertices of the pentagon to every other vertex.

Step 4. Draw a line from each vertex of the large pentagon to the opposite vertex of the

small pentagon.

Step 5. Using a pen, highlight the lines to be kept.

Step 6. Erase the remaining pencil lines and colour in your finished design.

THINK DRAW

Using a pair of compasses, draw a circle with

a 1 cm radius in the middle of your page.

Mark the centre of the circle.

To draw a pentagon in the circle, place the

centre of the protractor at the centre of the

circle and mark off every 72o along the

circumference. Join your markings with

straight-line segments.

Join each vertex of the pentagon to every

other vertex with straight lines.

Join each vertex of the large pentagon to the

opposite vertex of the small pentagon with

straight lines. Using a pen, highlight the lines

to be kept, so that the star is formed.

Erase the original circle and all the pencil

lines that have not been highlighted. Colour

in your finished design.

1

2

3

4

5

25WORKEDExample

Always work with a pencil first. When the constructions are done, use a pen to

highlight the lines to be kept and erase the remaining pencil lines.

remember


Geometric constructions

1 Use the steps outlined below to construct a larger version of this hexagonal pattern..

a Draw a circle of radius 10 cm.

b Draw a hexagon (mark off every 60° on the circle first).

c Mark the midpoint of each side of the hexagon.

d Join each midpoint to all other midpoints.

e Draw lines joining all vertices of the larger hexagon to each other (see the guide-

lines above).

f Using a pen, highlight the lines to be kept.

g Erase the remaining pencil lines and colour in your finished design.

2 Use the steps outlined below to construct the optical illusion as shown here in miniature.



c Join all vertices of the hexagon to each other.

d Draw a small hexagon in the middle of the diagram (see the guidelines above).

e Using a pen, highlight the lines to be kept.

f Erase the remaining pencil lines and colour in your finished design.

3 Construct the design, called Cubes using the steps outlined. Try different colouring

patterns.

8JWORKED

Example

25SkillSH

EET 8.13

Using a pair of compasses to draw circles Guidelines for step e Completed pattern, step g

Guidelines, Completed illusion,

step fstep d

Guidelines Final pattern




c Mark the midpoint of each side of the hexagon.

d Draw in the gridlines as shown in the guidelines above.

e Using a pen, highlight the lines to be kept.

f Erase the remaining pencil lines and colour in your finished design.

4 By following the steps outlined below, construct the Olympic rings.

Guidelines

a Draw a line 16 cm long in the centre of a page.

b Mark three points at 0 cm, 8 cm and 16 cm on this line. Rub out the line.

c Using each of these points as a centre, draw three circles of radius 5 cm.

d Taking the bottom points of intersection of these circles, draw another two circles of

radius 5 cm using the points of intersection as the centres as shown in the guidelines

above.

e Using the same five points as centres, draw five circles of radius 3 cm.

f Colour to create overlapping Olympic rings.

Nets and solids

A net of a solid is a 2-dimensional plan that can be cut out and folded to form that

solid.

Most packaging boxes are constructed from nets. This box which contained paperclips

can be unfolded to form a net as shown below.


Technology and polyhedraA demonstration version of the program Poly is available on the Maths Quest 8

CD-ROM. This program allows you to visualise polyhedra and their nets.

Click here on the CD-ROM for further instructions on how to install Poly.

Nets and solids

1 Draw and cut out the net shown and then

fold it into a tetrahedron.

a Make a net for a cube of side length 5 cm. Include flaps or tabs to hold the cube

securely together.

b Construct the box from your net.

THINK DRAW/CONSTRUCT

a A cube has six faces, each of which

is a square. Arrange six squares of

side 5 cm into a net.

Note: These figures have been

reduced to fit.

a

Add tabs to hold the cube together.

(This may take some trial and error.)

b Cut out the net and fold to crease all

the lines.

b

Fold into shape and stick down the

tabs with tape or glue.

1

2

1

2

26WORKEDExample

Poly

1. A net is a 2-dimensional plan that can be folded to create a 3-dimensional

object.

2. When designing nets, think carefully about placing tabs to give strength to your

solid.

3. When cutting out nets, accuracy is important.

remember

8K


2 Cut out the net shown and

fold it into an octahedron.

Stick down the tabs to make

your figure more stable.

3 a A cereal box measures 24 cm × 16 cm × 6 cm. Make a

net for this box. Include flaps or tabs to hold the box

securely together.


4 a Use cardboard to make a net for this chocolate box below.

Include flaps or tabs to hold the box securely together.


5 a Many people store pamphlets or magazines in pamphlet boxes

like the one shown. Use cardboard to make a net for this box.

Include flaps or tabs to hold the box securely together.


1 Find two different boxes at home (such as a milk carton, pizza box or other

packaging). Open up all the tabs and lay the boxes flat to make nets.

a Look at the way that the tabs are used to hold the boxes together. How strong

are the boxes?

b The net of each box was probably cut from a rectangle. Was much cardboard

wasted to make this net? Estimate the amount wasted and explain how you

have arrived at this estimation.

2 Make a net for a box with a 7 cm square base and height

10 cm. It should hold together without any tape or glue,

and have a closeable lid. Construct the box from

cardboard to check that it works.

WORKED

Example

26

WorkS

HEET 8.3

DESIGN Box it!

10 cm

7 cm7 cm


1 What types of angles can you identify in this

picture?

2 What types of triangles can you identify in this

picture?

3 What types of quadrilaterals can you identify in this

picture?

4 What other 2-dimensional shapes can you identify

in this picture?

5 Where are there any parallel lines in the picture?

6 Find two alternate angles in the diagram and

measure their sizes with a protractor. Are they

equal?

7 Find two corresponding angles in the diagram and


equal?

8 Find two co-interior angles in the diagram and


supplementary?

9 Does the Harbour Bridge have any axes of

symmetry?

Internet research

Use the Internet to see if you can answer the following

questions about the Sydney Harbour Bridge.

1 How long did it take to build the Sydney Harbour

Bridge? In what year was the Sydney Harbour

Bridge opened?

2 How tall are the pylons at either end?

3 How many rivets were used in the construction of

the bridge?

4 What is the average gap between the sea level and

the bottom of the bridge?

5 How high is the top of the arch above sea level?

6 The bridge is made of steel, which means that it

will expand and contract as the weather gets hotter

or colder. The builders made allowances for this.

How far can the archway expand? How far can the

deck of the bridge expand?

THINKING Geometry and the Sydney Harbour Bridge


Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows.

1 An triangle has all 3 sides of equal length and all 3 angles equal.

2 An isosceles triangle has of equal length and 2 base angles equal.

3 All sides and all angles are different in triangles.

4 An acute-angled triangle has all angles 90°.

5 An obtuse-angled triangle contains one angle.

6 A triangle contains one 90° angle.

7 The sum of the angles in any triangle is 180°.

8 An of a triangle is equal to the sum of the 2 interior anglesnot adjacent to it.

9 All can be divided into 2 groups: parallelograms and others.

10 Parallelograms have 2 pairs of and include rectangles,squares, parallelograms and rhombuses.

11 A rectangle has 2 pairs of opposite sides equal and all 4 angles are angles.

12 A has all 4 sides of equal length and four right angles.

13 A parallelogram has two pairs of of equal length and oppo-site angles of equal size.

14 A rhombus has sides of equal length and opposite angles of equalsize.

15 Other quadrilaterals include kites, and irregular quadrilaterals.

16 The of a kite are equal in length and the angles between theunequal sides are equal in size.

17 A trapezium is a quadrilateral with of parallel sides.

summary


18 An quadrilateral does not have any special features.

19 The sum of the interior angles in any quadrilateral is .

20 The sum of the interior angles in any polygon equals 180° × (n − 2), wheren is the in the polygon.

21 A polygon has all sides of equal length and all angles of equal size.

22 A is a line that intersects a pair (or a set) of parallel lines.

23 Corresponding angles (F-shaped), angles (Z-shaped) and angles (X-shaped) are equal in size.

24 Co-interior angles are (that is, they add to 180°).

25 If the length of each of the 3 sides of a triangle is known, it can be con-structed using a and a ruler.

26 If the length of 2 sides of a triangle and the size of the included angle, orthe size of the 2 angles and the length of the side between them areknown, a triangle can be constructed with the aid of a ruler and a .

27 A 2-dimensional picture of a 3-dimensional object is called an of that object.

28 In isometric drawings, vertical edges of an object are shown with verticallines, and the lines represent horizontal edges of an object.

29 A is a 2-dimensional plan, which can be cut and folded to form anobject.

W O R D L I S T

trapeziums

regular

isometric view

equilateral

parallel sides

protractor

alternate

two sides

scalene

360°

smaller than

square

obtuse

right-angled

all four

net of an object

transversal

supplementary

interior

adjacent sides

right

exterior angle

quadrilaterals

at an angle

verticallyopposite

opposite sides

one pair

irregular

number of sides

pair ofcompasses


1 Give: i the side name and ii the angle name for each of these triangles.

a b c

2 Using side and angle markings, draw a triangle that is:

a equilateral b right-angled and isosceles c obtuse-angled and scalene.

3 Find the values of the pronumerals in these triangles.

a b c d e

4 A ladder meets with the wall at a 30° angle. Find the angle that the ladder makes with the

ground.

5 Maya measures the angles in a triangle and finds that two angles are the same and one is 30°

larger than the other two. What is the size of each angle?

6 Calculate the value of the pronumeral in each diagram below.

a b c d

7 The slice of watermelon at right is to be cut into an isosceles

triangle. Find the size of each of the unknown angles.

8 Draw these quadrilaterals, showing all parallel sides, sides of

equal length and angles of equal sides, using appropriate

markings:

a a rectangle b a trapezium

c a kite d a rhombus.

8A

CHAPTERreview

8A

8B

65°

42°t50°

m

30°

w

40°

3x x

2x

2x + 10

8B

8C

50 t

62° x

50°

120°y 80°

2x

6x

65º

e

f

d

8C

8D


9 Find the value of the pronumeral in each diagram below.

a b c d

10 A kite has a bottom angle of 50° and a top angle of 120°.

Find the size of the other two angles of the kite.

11 Find the angle sum of:

a a pentagon b an octagon c a decagon.

12 Calculate the value of the pronumeral in each diagram below.

a b c

13 Calculate the size of each angle (labelled p) in this

regular pentagon.

14

a In the following diagrams, angles a and b are:

A vertically opposite

B corresponding

C co-interior

D alternate

E supplementary

b A vertically opposite

B corresponding

C co-interior

D alternate

E supplementary

8E

g55°

125° 125° x

110°

50°

2x

x

2x

x

x + 50°

x + 10°

8E

120°

50°

8F

8F

x

120°

130°

75º

210º

210º

75º

m

2m

2m

3m4m

m

p p

p

p

p

8G

8G multiple choice

a

b

a

b


c A vertically opposite

B corresponding

C co-interior

D alternate

E supplementary

d A vertically opposite

B corresponding

C co-interior

D alternate

E supplementary

15 For each diagram, calculate the value of the pronumeral and state the type of angle relation

that you used.

a b

c d

16 Use a pair of compasses, protractor and ruler to construct these triangles:

a a triangle with the side lengths 4 cm, 5 cm and 6 cm

b a triangle with two of the sides 4 cm and 5 cm long, and an angle between them of 75°.

17 Draw the front, right and top views of these objects.

a b

18 Draw isometric views of the objects, whose front, right and top views are given below.

a b

a

b

a

b

8G

60°

x

135°

45°

y

130°

t

68º

130ºe

c

a

b

d

8H

8I

8I

F

R

T

F R T


19 A rectangular prism, constructed from the set of cubes, is 3 cubes long, 2 cubes wide and

4 cubes high. Draw an isometric view of the prism.

20 Construct a decagonal star based on the

figures at right and using these steps.


b Mark the vertices of a decagon

(36° apart) on the circle. Also

mark the centre of the circle.

c Draw the lines shown in the guide at

right.

d Using a pen, highlight the lines to be

kept.

e Erase the remaining pencil lines and colour in your design.

21 Draw a net of the milk carton shown below.

22 This chocolate box has a 100 cm2 square base.

The height is 2.5 cm. Construct a possible net.

8I

Guidelines

8J

testtest

CHAPTER

yourselfyourself

8

8K

8K

Chap 08

Documents

Transcript of Chap 08